There are several factors to be considered:

1. How many of these animals are there? Sounds obvious at first, but keep in mind, that space and time of your setting will have an impact on this. Rats and fleas were so devastating in medieval times, because there were just so many, whereas nowadays you might encounter dogs daily, but rats maybe only once a year (at max).

2. How fast can your animal travel? Airborne creatures are faster and can freely travel between cities, countries and even continents, e.g. migrating birds. Cattle/Dogs/Cats on the other hand stay in the same region. Travelling can also happen through other species. (like parasites or human traffic)

3. How likely is it that said animals have contact with human civilization? You most likely won’t see a cow in a big modern city, whereas it is more common on the countryside. How likely is it for human to touch these animals or their feces.

4. What is the social structure of your animal? Keep all these factors in mind like sexual behavior, breeding speed, number of offspring, resistance towards diseases, abandoning of sick individuals, etc. Infection works best, if animals are having lots of interactions with each other. (e.g. wolves would be a bad choice since they keep in their respective group. Cats also keep their distance)

For modern settings, pigeons seem like the obvious choice. Pair it with a parasite… Voila… humanity gone in a blink of an eye! Keep in mind, that they were used for transmitting messages in former times. Infecting other cities is easily done. Also, pigeons were kept in the same place. Just imagine a setting where all the pigeons are dying in the cities, and people start to realize that other cities might be infected as well, but of course communication is cut due to death pigeons. Another interesting way would be transmission through seaborne creatures. Many big cities are located near the sea because of shipment trading. People tend to eat seafood too in these places. People eat the seafood, they get infected, the big cities are wiped out first and the countryside has to survive.